Waiting in Line Queuing theory (or waiting-line theory)...

Waiting in Line Queuing theory (or waiting-line theory) investigates the problem of providing adequate service economically to customers waiting in line. Suppose customers arrive at a fast-food service window at the rate of 9 people per hour. With reasonable assumptions, the average time (in hours) that a customer will wait in line before being served is modeled by the rational function $$f(x)=\frac{9}{x(x-9)}$$ where $x$ is the average number of people served per hour. A graph of $f(x)$ for $x>9$ is shown in the figure. (a) Why is the function meaningless if the average number of people served per hour is less than $9 ?$ Suppose the average time to serve a customer is 5 min. (b) How many customers can be served in an hour? (c) How many minutes will a customer have to wait in line (on the average)? (d) Suppose we want to halve the average waiting time to $7.5 \mathrm{~min}\left(\frac{1}{8} \mathrm{hr}\right)$. How fast must an employee work to serve a customer (on the average)? (Hint: Let $f(x)=\frac{1}{8}$ and solve the equation for $x$. Convert the answer to minutes and round to the nearest hundredth.) How might this reduction in serving time be accomplished?

Waiting in Line Queuing theory (or waiting-line theory) investigates the problem of providing adequate service economically to customers waiting in line. Suppose customers arrive at a fast-food service window at the rate of 9 people per hour. With reasonable assumptions, the average time (in hours) that a customer will wait in line before being served is modeled by the rational function $$f(x)=\frac{9}{x(x-9)}$$ where $x$ is the average number of people served per hour. A graph of $f(x)$ for $x>9$ is shown in the figure.
(a) Why is the function meaningless if the average number of people served per hour is less than $9 ?$
Suppose the average time to serve a customer is 5 min.
(b) How many customers can be served in an hour?
(c) How many minutes will a customer have to wait in line (on the average)?
(d) Suppose we want to halve the average waiting time to $7.5 \mathrm{~min}\left(\frac{1}{8} \mathrm{hr}\right)$. How fast must an employee work to serve a customer (on the average)? (Hint: Let $f(x)=\frac{1}{8}$ and solve the equation for $x$. Convert the answer to minutes and round to the nearest hundredth.) How might this reduction in serving time be accomplished?

Key Concepts

Arrival Rate

The arrival rate is the measure of how many customers or items enter a system per unit of time. It is a critical parameter in queuing theory since the balance between the arrival rate and the service rate determines if a system will operate smoothly or become congested. A higher arrival rate without a corresponding increase in service capacity can lead to excessive waiting times and unstable queue lengths.

Service Rate

The service rate is the average number of customers or items that can be served by a system per unit of time. In queuing models, the service rate must generally exceed the arrival rate for the system to be stable; otherwise, customers will accumulate faster than they are served, leading to infinite queues over time. Understanding and optimizing the service rate is essential in minimizing waiting times and ensuring smooth operation.

System Stability

System stability in queuing theory refers to the condition where the service rate is higher than the arrival rate, preventing the indefinite build-up of waiting customers. If the average number of people served per hour is less than the rate at which customers arrive, the queue will grow without bound, making the system unsustainable. Stability is a key consideration when designing service systems to ensure that they meet both customer needs and economic constraints.

Rational Function Modeling

Rational functions are often used in queuing theory to model relationships between key parameters such as service rate and waiting time. Such functions capture the non-linear behavior as system parameters change, particularly near critical thresholds. In this context, the given rational function helps quantify how changes in the service rate directly affect the average waiting time, providing a mathematical tool to analyze and optimize service efficiency.

Queuing Theory

Queuing theory is the branch of operations research that studies the behavior and performance of waiting lines. It involves analyzing models that help predict system performance metrics such as average wait times and the number of customers in line based on parameters like arrival rates and service rates. This understanding is crucial for designing efficient service systems where customer satisfaction and cost effectiveness are priorities.

Transcript

00:02 Queuing theory, which is also known as waiting line theory, looks at the problem of providing adequate service economically to customers that are waiting in line.

00:11 So suppose that customers arrive at a fast food service window at the rate of nine people per hour.

00:17 With reasonable assumptions, the average time in hours that a customer will wait in line before being served as modeled by this function.

00:24 F of x equals 9 over x times x minus 9.

00:30 X is the average number of people served per hour.

00:34 And this graph here is the graph of f of x for x is greater than nine.

00:38 So the first thing we want to look at is one.

00:42 Why is the function meaningless if the average number of people served per hours less than nine? so the reason is because the function, so if it was less than nine, because the function would be negative.

01:14 So because the function is negative for the average number of people served per hour, if it's less than nine.

01:30 And we can see that by plugging in a number less than nine.

01:33 So let's say x was one.

01:35 Inside the parentheses, we get one minus nine, which is negative eight.

01:41 And i would have negative eight times one, which is negative eight.

01:44 And now my whole thing is negative.

01:46 And that doesn't make sense.

01:49 The second thing we're gonna look at is how many customers can be served in an hour.

01:54 So to figure this out, we're gonna take 60.

01:56 We're gonna divide it by five because we are going to suppose the average time to serve a customer is five minutes.

02:03 So that means every five minutes, they're serving a new customer.

02:06 They're 60 minutes in an hour.

02:07 So we're going to say 60 divided by five.

02:10 And we get that they would serve 12 people per hour.

02:17 Third, we're going to look at is still supposing the average time to serve a customer is five minutes.

02:24 How many minutes will customer have to wait in line on average? so to figure this out, remember we're serving 12 people per hour...

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